The Modern Mathematics of Deep Learning

An entropy integral

Penguin learning project

The project uses 342 complete Palmer penguin records: 240 training-pool and 102 held-out rows. The selected training rows determine feature scaling and model parameters. Source CSV

This project trains weights with softmax, unlike the homepage lab, which uses nearest species averages. Same penguin measurements; different classification methods. Predictions can differ.

Generalization and risk bounds · Relate sample behavior to population risk under stated assumptions.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

Consider this concept alongside the project; its source assumptions may differ. These observations do not prove a theorem.

Input: 12 × 2; Held-out evaluation: 102 · #58 · 40.6 mm, 193 mm. Known answer: Adélie; Prediction: Adélie · [≈0.7383, ≈0.2444, ≈0.01737].

Used in these source sections: 1.2 Learning from examples

The source relates Rademacher complexity to an integral of the logarithm of a covering number over radius α.

For the three constant functions 0, 1/2 and 1, the open-ball covering number is 3 for 0 < α ≤ 1/2 and 1 for α > 1/2.

Covering numbers summarize function-class complexity at many scales.

Source formula example

{\mathfrak{R}}_{m}({\mathcal{G}})\lesssim{\mathbbm{E}}\Big[\int_{0}^{\infty}\sqrt{\frac{\log N_{\alpha}({\mathcal{G}},d_{S})}{m}}\,\mathrm{d}\alpha\Big],

illustration · This is an analytically integrated finite class, not a proof. The symbol ≲ hides a constant, so the displayed integral alone is not a numerical upper bound with constant 1.

Symbols

$N_α$: Number of open sample-distance balls needed to cover the class.
$m$: Number of sample points.

Source formula example

This source example uses its own inputs and assumptions. Editing it does not change the project.

Edit a value or the code, then run the example. This is a finite demonstration, not a proof of the paper’s theorem.

Radius α

This example’s data flow

You choose

input.radius = 0.25

Initial values in code

m = 2
signs = [ [-1, -1], [-1, 1], [1, -1], [1, 1], ]

Calculated here

Result expression: { coveringNumber, integral, complexity }

coveringNumber = input.radius <= 0.5 ? 3 : 1 Uses: input.radius
integral = 0.5 * Math.sqrt(Math.log(3) / m) Uses: m
complexity = signs.reduce((sum, pair) => sum + Math.abs((pair[0] + pair[1]) / 2), 0) / signs.length Uses: signs

Returns

Example result

coveringNumber: 3
integral: ≈0.3706
complexity: 0.5

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

The sample distance between these constant functions is their absolute value difference, so the covering number is independent of the sample. The source expectation is therefore this fixed integral for the toy. At the boundary α=1/2 the balls are open; changing one endpoint does not change the integral. Random-sign complexity is computed by enumerating all four equally likely sign pairs.

Source: Source expression